Quantum key exchange (only one of the applications of "quantum internet") is completely orthogonal to post quantum encryption.
Quantum key exchange is a way to distribute keys, which will be used in some symmetric encryption scheme that runs on classical computers. That scheme has security that does not depend on any "difficulty" conjectures. It is "information theoretically perfect" modulo implementation bugs. (Assuming Schroedinger's equation is not wrong).
This does not require the creation of a quantum computer. Actually it is much easier and already done in labs.
Post quantum encryption is just the name for asymmetric encryption algorithms that run on classical computers and are difficult to break on both classical and quantum computers.
Even though quantum key exchange is an exciting application of "quantum internet", way more interesting would be its use for the establishment of globe spanning entangled states. This can be used for the creation of sensors that surpass the diffraction limit and other practical "science stuff".